Cell Graphs: A Provable Correct Method for the Storage of Geometry *

Several methods have been proposed for the storage of geometric properties in Geographic Information Systems, but many are based on the storage of metric data (coordinates) and analytical geometry. Because of the well-known limitations of implementations of the algebra of computer real numbers, such as incidence and inclusion during affine transformations. We propose an approach in which topological relations are separately recorded and independent of metric positions. The method is based on the use of simplices, which are the simplest polyhedrons of each dimension. The zero- dimensional simplex is the point, the one-dimensional one line, etc. In order to allow for non-straight lines as connections between points, we actually use cells, which are the homeomorphic image of simplices. In order to store topological relations, we use two completeness principles: Completeness of incidence and completeness of inclusion. We can show that in such a geometric configuration topology is invariant to affine transformations, independently of the method selected for recording metric information. For formal treatment we form a multi-sorted algebra (abstract data types). The axioms for this algebra must be selected such that the above-mentioned principles are maintained as invariants. We rely on an arbitrary method to learn about the topological relations initially. This "oracle" may use the calculation of a distance and a threshold, query the user or decide randomly, but it cannot influence the consistancy of the resulting geometry, as it is consulted only if the same information was not previously available and thus cannot lead to an inconsistent situation. Reasonable performance is expected, as this method imposes a 'neighborhood' structure on the data. All operations use and change only data of objects in immediate proximity. Databases suitable for handling spatial data should permit clustering of data by proximity.